Abstract
The complexity and decidability of various decision problems involving the shuffle operation (denoted by 
) are studied. The following three problems are all shown to be \(\mathsf{NP}\)-complete: given a nondeterministic finite automaton (\(\mathsf{NFA}\)) \(M\), and two words \(u\) and \(v\), is 
, is 
, and is 
? It is also shown that there is a polynomial-time algorithm to determine, for \(\mathsf{NFA}\)s \(M_1, M_2\) and a deterministic pushdown automaton \(M_3\), whether 
. The same is true when \(M_1, M_2,M_3\) are one-way nondeterministic \(l\)-reversal-bounded \(k\)-counter machines, with \(M_3\) being deterministic. Other decidability and complexity results are presented for testing whether given languages \(L_1, L_2\) and \(L\) from various languages families satisfy 
.
The research of O. H. Ibarra was supported, in part, by NSF Grant CCF-1117708.
The research of I. McQuillan was supported, in part, by the Natural Sciences and Engineering Research Council of Canada.
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Eremondi, J., Ibarra, O.H., McQuillan, I. (2015). On the Complexity and Decidability of Some Problems Involving Shuffle. In: Shallit, J., Okhotin, A. (eds) Descriptional Complexity of Formal Systems. DCFS 2015. Lecture Notes in Computer Science(), vol 9118. Springer, Cham. https://doi.org/10.1007/978-3-319-19225-3_9
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